This result improves the bounds for the case of general constraints of the previous section, and yields only \(1/4\)  worse constraint violation than the bounds in [1]} – which however cannot benefit from predictions. On the contrary, we see that with LLP, due to its prediction-adaptive steps, we get no regret for perfect predictions and, interestingly, the constant factors of the bounds shrink commensurately with the predictions' accuracy. This becomes clear if we express the bounds as:
\(&\!\!\mathcal {R}_T\!\le B_T\!\triangleq \!A_1\sqrt{h_{1:T}}+\min \left\lbrace \! 2a\sqrt{\sum _{t=1}^T\xi _t^2 },\, A_2T^{1-\beta } \right\rbrace   \\&\mathcal {V}_T\!\le \left[ A_3\max \left\lbrace  K_T, T^\beta \right\rbrace \big (B_T-R_T\big ) \right]^{\frac{1}{2}} +A_4 \sqrt{h_{1:T}} \) 
